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Q.

There are $5$ letters and $5$ directed envelopes. The number of ways in which all the letters can be put in wrong envelope is ____

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Detailed Solution

Here, we have to realize that no letter is placed in the appropriate envelope. To do this, total up the different ways that each letter can be placed in the appropriate envelope. This means that only one letter can fit in one proper envelope, and you can subtract them all from the total number of ways without any restrictions, regardless of whether you're looking for the right or wrong address.
5 letters 5 directed envelope
All the of letters Put wrong wrong :
Number of ways =

[1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}] 5!

=[

1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120}] 5!

=

[\frac{2}{6} + \frac{4}{120}] 5!

= [\frac{1}{3} + \frac{1}{30}] 5!

=

[\frac{11}{30}] 120

=44
Hence, option 2 is correct

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There are 5 letters and 5 directed envelopes. The number of ways in which all the letters can be put in wrong envelope is ____